Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $n = \dfrac{2p}{7(3p - 10)} \div \dfrac{3p}{12p - 40} $
Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{2p}{7(3p - 10)} \times \dfrac{12p - 40}{3p} $ When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 2p \times (12p - 40) } { 7(3p - 10) \times 3p } $ $ n = \dfrac {2p \times 4(3p - 10)} {3p \times 7(3p - 10)} $ $ n = \dfrac{8p(3p - 10)}{21p(3p - 10)} $ We can cancel the $3p - 10$ so long as $3p - 10 \neq 0$ Therefore $p \neq \dfrac{10}{3}$ $n = \dfrac{8p \cancel{(3p - 10})}{21p \cancel{(3p - 10)}} = \dfrac{8p}{21p} = \dfrac{8}{21} $